Finite Group Databasebeta

← #13154778

Subgroups — #13154778

Order 1728.

360 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723435C211yesmaximal, characteristic
2#11723103C211yesmaximal, characteristic
3C3×#63781C211yesmaximal, characteristic
4#10590215—31nomaximal
5C22×C3×#157—21no—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C32×Dic6C411yescharacteristic, complemented
8C2×C3⋊#622—21no—
9C4⋊(C3⋊(C3⋊Dic3))C411yescharacteristic, complemented
10#88806—21no—
11#88803—21no—
12#64155—31no—
13#63856—62no—
14#63781D311yescharacteristic
15#63778—31no—
16C22×C3×C3⋊D3D411yescharacteristic
17C2×C3×#157—22no—
18C2×C3×#157—41no—
19C2×C3×#157—41no—
20C2×C33×C4C2×C411yescharacteristic
21C2×C3⋊(C3⋊Dic3)—21no—
22C2×C3×C3⋊Dic3D411yescharacteristic
23C3×C4×C3⋊D3—41no—
24C2×C32×Dic3—21no—
25C32×Dic6—41no—
26C3⋊#622—41no—
27#4452—91nomaximal
28C22×#157—21no—
29C2×C32×Q8—31no—
30C2×C4×C3⋊D3C2×D311yescharacteristic
31C2×C3×C4×D3—62no—
32C2×C3×Dic6—22no—
33C2×#622—61no—
34#3752—61no—
35C4⋊(C3⋊Dic3)—31no—
36C4⋊(C3⋊Dic3)—62no—
37C4⋊(C3⋊Dic3)—122no—
38#3750—61no—
39#3724—64no—
40(C2×D3)⋊Dic3—64no—
41C22×C33C22⋊C411yescharacteristic
42C2×C3×C3⋊D3—21no—
43C2×C3×C3⋊D3—21no—
44C2×C3×C3⋊D3SD1611yescharacteristic
45C2×C3×C3⋊D3Q1611yescharacteristic
46C3×#157—42no—
47C33×C4—21no—
48C3⋊(C3⋊Dic3)—41no—
49C3×C3⋊Dic3—21no—
50C32×Dic3—41no—
51#946—182no—
52#924—91no—
53C3×#95—91no—
54#830—91no—
55C22×C3⋊D3#6711yescharacteristic
56C22×C3×D3—62no—
57C2×#157—22no—
58C2×#157—41no—
59C2×#157—41no—
60C32×Q8—121no—
61C2×C32×C4—22no—
62C2×C32×C4—42no—
63C2×C32×C4—61no—
64C2×C32×C4C4×D311yescharacteristic
65C2×C3⋊Dic3—61no—
66C2×C3⋊Dic3—64no—
67C2×C3⋊Dic3—124no—
68C2×C3⋊Dic3D1211yescharacteristic
69C4×C3⋊D3—41no—
70C2×C3×Dic3—24no—
71C2×C3×Dic3—62no—
72C3×C4×D3—124no—
73C3×Dic6—44no—
74#622—121no—
75#328—271noSylow
76C2×C33#9911yescharacteristic
77C2×C33#10311yescharacteristic
78C2×C33#10411yescharacteristic
79C3×C3⋊D3—21no—
80C3×C3⋊D3—21no—
81C2×C3×Q8—62no—
82C22×C3×C4—91no—
83C22×C3×C4—181no—
84C2×C4×D3—62no—
85C2×Dic6#15711yescharacteristic, complemented
86#217—181no—
87(C2×D3)⋊C4—184no—
88C4⋊Dic3—91no—
89C4⋊Dic3—182no—
90C4⋊Dic3—362no—
91Dic3⋊C4—181no—
92Dic3⋊C4—184no—
93C2×C3⋊C8—181no—
94C22×C32—22no—
95C22×C32—42no—
96C22×C32(C2×D3)⋊C411yescharacteristic
97C2×C3⋊D3—21no—
98C2×C3⋊D3—21no—
99C2×C3⋊D3#21411yescharacteristic
100C2×C3⋊D3#21611yescharacteristic
101C2×C3×D3—62no—
102C2×C3×D3—62no—
103C2×C3×D3—122no—
104C2×C3×D3—122no—
105#157—42no—
106C32×C4—21no—
107C32×C4—24no—
108C32×C4—44no—
109C32×C4—121no—
110C3⋊Dic3—21no—
111C3⋊Dic3—121no—
112C3⋊Dic3—124no—
113C3⋊Dic3—244no—
114C3×Dic3—44no—
115C3×Dic3—122no—
116C22⋊Q8—271no—
117C22⋊C8—271no—
118#95—91no—
119C33#32811yescharacteristic, complemented, Sylow
120C23×C3—91no—
121C22×D3—62no—
122C3×Q8—124no—
123C2×C3×C4—22no—
124C2×C3×C4—42no—
125C2×C3×C4—64no—
126C2×C3×C4—91no—
127C2×C3×C4—182no—
128C2×C3×C4—361no—
129C2×C3×C4—361no—
130C2×C3×C4—361no—
131C2×C3×C4C2×#15711yescharacteristic
132C2×Dic3—21no—
133C2×Dic3—62no—
134C2×Dic3—184no—
135C2×Dic3—181no—
136C2×Dic3—364no—
137C4×D3—124no—
138Dic6—41no—
139C3⋊C8—361no—
140C2×C32—22no—
141C2×C32—22no—
142C2×C32—22no—
143C2×C32—42no—
144C2×C32—42no—
145C2×C32—42no—
146C2×C32#80511yescharacteristic
147C2×C32#80611yescharacteristic
148C2×C32#81011yescharacteristic
149C3⋊D3—21no—
150C3⋊D3—21no—
151C3×D3—122no—
152C3×D3—122no—
153C2×Q8—31no—
154C22×C4—91no—
155C22×C4—181no—
156C2×C8—541no—
157C4⋊C4—271no—
158C4⋊C4—541no—
159C22⋊C4—541no—
160C22×C3—22no—
161C22×C3—42no—
162C22×C3—91no—
163C22×C3—91no—
164C22×C3—181no—
165C22×C3—181no—
166C22×C3C22⋊#15711yescharacteristic
167C2×D3—62no—
168C2×D3—62no—
169C2×D3—122no—
170C2×D3—122no—
171C3×C4—24no—
172C3×C4—21no—
173C3×C4—44no—
174C3×C4—124no—
175C3×C4—181no—
176C3×C4—362no—
177Dic3—41no—
178Dic3—122no—
179Dic3—364no—
180Dic3—361no—
181Dic3—724no—
182C32—22no—
183C32—42no—
184C32#445211yescharacteristic, complemented
185C23—91no—
186Q8—121no—
187C2×C4—61no—
188C2×C4—91no—
189C2×C4—182no—
190C2×C4—361no—
191C2×C4—361no—
192C2×C4—361no—
193C2×C4—541no—
194C2×C4D3×#15711yescharacteristic
195C8—1081no—
196C2×C3—22no—
197C2×C3—22no—
198C2×C3—22no—
199C2×C3—42no—
200C2×C3—42no—
201C2×C3—42no—
202C2×C3—181no—
203C2×C3—181no—
204C2×C3#6378411yescharacteristic
205C2×C3#6378511yescharacteristic
206C2×C3#6378611yescharacteristic
207D3—122no—
208D3—122no—
209C22—91no—
210C22—91no—
211C22—181no—
212C22—181no—
213C22(C2×D3)⋊#15711yescharacteristic
214C4—21no—
215C4—121no—
216C4—181no—
217C4—362no—
218C4—1081no—
219C3—22no—
220C3—42no—
221C3#1059021511yescharacteristic, complemented
222C2—181no—
223C2—181no—
224C2#1172270811yescharacteristic
225C2#1172271311yescharacteristic
226C2#1172272711yescharacteristic
subgroupG/H
centerC22(C2×D3)⋊#157
commutator, FittingC2×C33×C4C2×C4
Frattini, hypercenterC2×C4D3×#157
socleC22×C33C22⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3×C4×C3⋊D3C22
hypocenterC33#328
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
7. C2×C32×Dic6C4semidirect
9. C4⋊(C3⋊(C3⋊Dic3))C4semidirect
85. C2×Dic6#157semidirect
119. C33#328semidirect
184. C32#4452semidirect
221. C3#10590215semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117234352
#117231032
C3×#637812
#105902153
#44529